Note: The comments/permalink issue on this post has been fixed. I have no idea exactly what happened, but it’s not happening anymore which is good enough for me!
I saw Pixar’s Wall-E on opening night. Since even the most mediocre Pixar films are usually among the best in the business, I figured it would be worth the seven bucks. I was not at all disappointed. It is a beautiful, beautiful film - possibly the studio’s best and most touching. I cannot possibly recommend it highly enough.
There is a lovely scene where the titular robot uses a fire extinguisher to propel himself through the vacuum of space. There’s no sense in critiquing the physics of a gentle animated film, but it gives us an opportunity to talk about the principal challenge of moving about in space - there’s nothing to push against. On earth you push against the ground with your feet while walking, or with your tires when driving. If you’re in an airplane, the propellers or jet engines pull in still air in front of the plane and push it out the back at high speed. Boats do the same thing with water. It’s just Newton’s laws in action.
In space there’s just blank vacuum. You can spin your tires and turbines, flap your wings, and swing your feet but you’ll just be flailing in place. If you want to push against something, you’ll have to bring it with you. This something is rocket fuel. The faster you push it out the back of you spacecraft the faster you’ll go, which is why rocket fuel is ignited and blasted out the back as fast as its fiery chemistry can take it. But in theory you could fling rocks out the back by hand and it would accelerate you forward just as surely - if much more slowly. The problem is that eventually you run out of fuel or rocks to fling. How fast will you be going when you run out? Let’s assume (or pick an appropriate frame of reference) that you’re at rest when you start. The total momentum of the spacecraft/fuel system is zero, and since there are no outside forces it will remain zero during the process of flinging small rocks (or fuel) out the back. If we call the mass of the spacecraft M and its velocity V, and we call the mass of the rock m and its velocity v we can figure out what V is after you fling a rock.
Solve for V and you’re set - if you’re flinging just one rock. Flinging more rocks complicates things - M keeps changing and you have to iterate over and over. Worse, you’ll have to do so infinitely many times if you’re using a more-or-less continuous substance like rocket fuel as your propellant. Maybe we can improve matters if we figure out a way to describe this with an integral. Let’s see… we can leave little v alone because that’s the speed of whatever we’re flinging out - it stays constant. We can call the little bits of what we’re flinging out dM, since they were part of the original spaceship + fuel mass. We can call the change in speed of our spacecraft dV. Just like the above equation, after a bit of rearrangement this leaves us with
And to get our total change in velocity V, we just integrate this over the change in mass.
Where the i and the f mean the initial mass (the spaceship and the fuel) and the final mass (just the spaceship, emptied of fuel). Doing the integral gives
The ratio is called, appropriately, the mass ratio. If your fueled spacecraft is twice the mass of the unfueled spacecraft, you’ll end up moving at log(2) times whatever speed you were throwing fuel out the back. It’s pretty clear that even ludicrously huge mass ratios won’t do much good since the natural log is such a slowly growing function. You’re more or less confined to have a maximum speed on the order of the exhaust speed no matter how much fuel you pack.
Are there ways to get very high exhaust speeds? Sure, there’s lots of ways although most of them are pretty limited at the moment. Those will make a good subject for another time.
Note from yesterday: Traffic to this site easily set a record yesterday when I posted about politics and religion on science blogs, with visits up from the average by about a factor of two. Either this means I’m right and people are happy to read someone disagreeing with the prevalence of those topics in science blogs, or I’m totally wrong and even mentioning those topics drives up interest! I’m sticking to my guns either way, and keeping those topics to a minimum here.